If the PC is able to boot from the Floppy as described the Bios is fine.
The problem is that for some reason it is not able to boot to the HD possibly because the bios is not recognising the HD correctly or because the file system on the HD is shot.
From the A: drive try running fdisk to check the current setup of the HD.
It could just be a case of needing to get into bios by selecting the correct key (or keys) to hit as you boot the PC. Many use F1 or F2 or F10 some use the Del button and some use a combination of keys. You should be able to check which this is either by watching for this to be displayed at the start of the boot sequence if not looking up the specs for the mobo will tell you how to get into the bios.
Just because you can't see the post section of the boot doesnt mean it is not running or that you cant access bios.
My bet on this now is that maybe the cmos battery has died which cleared the bios settings back to defaults which fail to see the HD correctly or possibly had incompatable settings blocking the IDE channels.
Either way this is actually sounding more likely to be a solvable problem.